Integrate the functions`(5x)/((x+1)(x^2+9)`

Explanation

To solve the integral \(\int \frac{5x}{(x+1)(x^2+9)} \, dx\), we will use the method of partial fractions. Here’s a step-by-step solution:

### Step 1: Set Up Partial Fraction Decomposition
We want to express the integrand \(\frac{5x}{(x+1)(x^2+9)}\) as a sum of simpler fractions. We can write:
\[ \frac{5x}{(x+1)(x^2+9)} = \frac{A}{x+1} + \frac{Bx + C}{x^2 + 9} \]
where \(A\), \(B\), and \(C\) are constants that we need to determine.
...